Problem: Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{z^2 - 25}{z + 5}$
First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = z$ $ b = \sqrt{25} = 5$ So we can rewrite the expression as: $p = \dfrac{({z} + {5})({z} {-5})} {z + 5} $ We can divide the numerator and denominator by $(z + 5)$ on condition that $z \neq -5$ Therefore $p = z - 5; z \neq -5$